Contraction in Wasserstein 1-distance with explicit rates is established for generalized Hamiltonian Monte Carlo with stochastic gradients under possibly nonconvex conditions. The algorithms considered include splitting schemes of kinetic Langevin diffusion. As consequence, quantitative Gaussian concentration bounds are provided for empirical averages. Convergence in Wasserstein 2-distance, total variation and relative entropy are also given, together with numerical bias estimates.

We consider dynamic pricing strategies in a streamed longitudinal data set-up where the objective is to maximize, over time, the cumulative profit across a large number of customer segments. We consider a dynamic model with the consumers' preferences as well as price sensitivity varying over time. Building on the well-known finding that consumers sharing similar characteristics act in similar ways, we consider a global shrinkage structure, which assumes that the consumers' preferences across the different segments can be well approximated by a spatial autoregressive (SAR) model. In such a streamed longitudinal set-up, we measure the performance of a dynamic pricing policy via regret, which is the expected revenue loss compared to a clairvoyant that knows the sequence of model parameters in advance. We propose a pricing policy based on penalized stochastic gradient descent (PSGD) and explicitly characterize its regret as functions of time, the temporal variability in the model parameters as well as the strength of the auto-correlation network structure spanning the varied customer segments. Our regret analysis results not only demonstrate asymptotic optimality of the proposed policy but also show that for policy planning it is essential to incorporate available structural information as policies based on unshrunken models are highly sub-optimal in the aforementioned set-up. We conduct simulation experiments across a wide range of regimes as well as real-world networks based studies and report encouraging performance for our proposed method.

While low-rank matrix prior has been exploited in dynamic MR image reconstruction and has obtained satisfying performance, tensor low-rank models have recently emerged as powerful alternative representations for three-dimensional dynamic MR datasets. In this paper, we introduce a novel deep unrolling network for dynamic MRI, namely the learned transform-based tensor low-rank network (LT$^2$LR-Net). First, we generalize the tensor singular value decomposition (t-SVD) into an arbitrary unitary transform-based version and subsequently propose the novel transformed tensor nuclear norm (TTNN). Then, we design a novel TTNN-based iterative optimization algorithm based on the alternating direction method of multipliers (ADMM) to exploit the tensor low-rank prior in the transformed domain. The corresponding iterative steps are unrolled into the proposed LT$^2$LR-Net, where the convolutional neural network (CNN) is incorporated to adaptively learn the transformation from the dynamic MR dataset for more robust and accurate tensor low-rank representations. Experimental results on the cardiac cine MR dataset demonstrate that the proposed framework can provide improved recovery results compared with the state-of-the-art methods.

We consider two-player zero-sum turn-based interactions, in which the sequence of possible successive moves is represented by a maximin game tree T. This tree models the possible actions sequences by a collection of MAX nodes, that correspond to states in the game in which player A should take action, MIN nodes, for states in the game in which player B should take action, and leaves which specify the payoff for player A. The goal is to determine the best action at the root for player A. For deterministic payoffs this search problem is primarily algorithmic, with several powerful pruning strategies available [20]. We look at problems with stochastic payoffs, which in addition present a major statistical challenge. Sequential identification questions in game trees with stochastic payoffs arise naturally as robust versions of bandit problems. They are also a core component of Monte Carlo tree search (MCTS) approaches for solving intractably large deterministic tree search problems, where an entire sub-tree is represented by a stochastic leaf in which randomized play-out and/or evaluations are performed [4]. A play-out consists in finishing the game with some simple, typically random, policy and observing the outcome for player A. For example, MCTS is used within the AlphaGo system [21], and the evaluation of a leaf position combines supervised learning and (smart) play-outs. While MCTS algorithms for Go have now reached expert human level, such algorithms remain very costly, in that many (expensive) leaf evaluations or play-outs are necessary to output the next action to be taken by the player. In this paper, we focus on the sample complexity of Monte-Carlo Tree Search methods, about which very little is known. For this purpose, we work under a simplified model for MCTS already studied by [22], and that generalizes the depth-two framework of [10].